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NBSIR 87-3572

Diffusion-Controlled Reaction in a

Vortex Field

Ronald G. Rehm, Howard R. Baum

and Daniel W. Lozier

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Applied Mathematics and

Center for Fire Research

Gaithersburg, MD 20899

June 1987

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

-QC

100

.U56

#87-3572

1987

r .9

Research Information Center

National Bureau of Stanaarda

Gaithersburg, JVlarykad 20B99

NBSIR 87-3572

DIFFUSION-CONTROLLED REACTION IN A

VORTEX FIELD

Ronald G. Rehm, Howard R. Baum

and Daniel W. Lozier

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Applied Mathematics and

Center for Fire Research

Gaithersburg, MD 20899

June 1987

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Diractor
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Diffusion- Controlled Reaction in a Vortex

Field

Ronald G. Rehm,* Howard R. Baum^ and Daniel W. Lozier^

National Bureau of Standards

Gaithersburg, MD 20899

June 4, 1987

Abstract

A two-dimensional model of a constant-density diffusion-controlled

reaction between unmixed species initially occupying adjacent halfspaces is formulated and analyzed. An axisymmetric viscous vortex

field satisfying the Navier-Stokes equations winds up the interface between the species as they diffuse together and react. A flame-sheet

approximation of the rapid reaction is made using Shvab-Zeldovich dependent variables. The model was originally proposed by F. Marble,

who performed a local analysis and determined the total consumption rate along the flame sheet. The present paper describes a global

similarity solution to the problem which is Fourier analyzed in a Lagrangian coordinate system. An asymptotic analysis of the Fourier

amplitudes, valid for large Schmidt numbers is presented. The solution is evaluated numerically in Lagrangian and Eulerian coordinate

systems. This problem has been studied as part of a more general

model which has application to the description of turbulent combustion.

* Center for Applied Mathematics

^Center for Fire Reseeirch

^Center for Applied Mathematics

1
1 Introduction

The theoretical study of chemical reactions in complex flow flelds, for ex- ample turbulent reacting flows, has received increased attention lately [l]-

[8j. This attention is warranted not only because numerical and analytical

progress is being made in addressing the problem, but also because a much

clearer physical picture of turbulent flows and their coupling to combustion

processes has been achieved through experiments over the past flfteen years.

[9]- [12]. Turbulent combustion is difficult to analyze because it is highly

nonlinear, transient and involves a wide range of length and time scales.

When the Reynolds number is large, experiments indicate that the length

and time scales associated with turbulent combustion can be separated by

phenomenon into large and small scales. The large, or geometrical, scale is essentially inviscid or nondissipative and is related to the geometry deflning

the flow conflguration and the fuel and oxidizer distributions. Combustion,

on the other hand, takes place on the small scale associated with the diffusion of fuel and oxidizer into each other; it “rides on” the geometrical

scale and establishes the rate at which the reactants disappear and heat is released.

An outline of a general approach for studying the problem of turbulent

reacting flows, based on these observations and using both analytical and

numerical methods, is given in a recent paper by Baum, Corley and Rehm

[13]. The approach is to analyze large-scale flow fields separately from the

small-scale mixing and reaction using the observation that these processes occur on widely differing length and time scales for large Reynolds numbers.

The component problems are treated individually, but in a way that will

allow the phenomena to be coupled through analytical and computational

techniques. Mathematical analysis is used extensively; however, the task is ultimately computational, and computations must be carried out for any

configuration of practical interest.

Also presented in [13] is a model for three-dimensional small-scale mixing and reaction in a stretched vortex flow field; this model includes the

critical features for turbulent combustion of flame stretching and the threedimensional effect of vortex stretching. In [14], Marble originally posed a

similar but more specialized two-dimensional model problem of small-scale

2
mixing and reaction and studied it analytically. His model includes the

two-dimensional effect of flame stretching, and was analyzed using methods presented earlier by Carrier, Fendell and Marble [15] locally along the

flame front to provide global dependences of the fuel and oxidizer con- sumption rates upon the governing parameters. The Marble problem has

several important features: it is a diffusion-controlled reaction in a viscously

spreading vorticity field which stretches the flame sheet. Even though it

is only two dimensional, it approximates chemical reactions in individual

vortices which occur in shear-layer mixing experiments, such as those of

Mungal and Dimotakis [llj. Generalizations of the Marble problem for a

stretched vortex flow field and for reactions which release heat, using a local analysis, have been performed by Karagozian [23] and Karagozian and

Marble [24]. We analyze the Marble problem in a manner very different

from that described in the original paper of Marble [14].

Two-dimensional numerical computations of the flow properties in re- acting mixing layers have recently been carried out [l6]-[20]. In these computations and the experiments they simulate, two fluids, one containing

fuel and the other containing oxidizer and each flowing unidirectionally

with its own velocity are brought into contact at a “splitter plate” . Downstream of the point of contact, the two fluids mix and chemical reactions

take place as they diffuse together. Experiments indicate that during the

early stages of development of these mixing layers, the flow fields remain

primarily two dimensional. Then, before vortex pairing begins, the Marble

problem can be regarded as an analytical approximation to combustion in

one of the vortices. Also, recently, a direct numerical computation of the

Marble problem has been undertaken by Laverdant and Candel [21]. All of

these computations demonstrate that, even for the small-scale problem of

mixing, diffusion and reaction, without attempting to calculate large-scale

flow fields also, the computations are large, complex and difficult.

The work presented in this paper is essentially analytical, and, within

the context of the mathematical model, permits one to calculate combustion

properties much more accurately for given computational resources than

direct numerical solution. Therefore, it could be used, for example, to test

the accuracy of the numerical computations cited above. The novel features

of our work compared with that of Marble [14] are (i) it is observed that

3
the convection-diffusion equation for the Shvab-Zeldovich variable permits a similarity solution, reducing the number of independent variables from

three (radius, angle and time) to two (angle and similarity variable); (ii) as in [13], a Lagrangian coordinate system is used to eliminate flame-sheet

resolution problems induced by vortex winding; (iii) Fourier analysis in

angle and a combined numerical and analytical treatment in the similarity

variable allow one to solve the global problem essentially exactly. The

solution to this problem will provide a special case on which to test methods

for solution to the more general problems described in [13]

In Section 2 we formulate the problem and give its complete mathematical description. Special cases of the general problem are presented

and solved in Section 3. In this section an approximate solution for large

Schmidt number is given which is particularly valuable because it displays

the analytical behavior of the problem. It is also appropriate for diffusion

controlled reactions in many miscible liquids where the Schmidt number

really is large. Results calculated both from the special cases and more

general numerical methods are given in Section 4. In Section 5 we calculate the global fuel and oxidizer consumption rates using the large Schmidt

number analytical result and assuming in addition that the Reynolds number is large. Finally, in Section 6, some conclusions are drawn from this

study.

2 Formulation of the Problem

Consider the model in which initially there is fuel in the left half-plane

and oxidizer in the right half-plane in arbitrary proportions. These halfspaces are brought into contact and simultaneously a line vortex with axis

at the origin is imposed (see Figure 1). The vortex induces a convective

mixing of the interface between the two species, increasing the area of the

separating surface in the neighborhood of the origin and enhancing the

diffusion of the species into each other. It is assumed that the reaction

rate is sufficiently rapid for the process to be limited by diffusion, and a

flame-sheet approximation is made for the reaction. The chemical reaction

is assumed to take place at constant density and all diffusion coefficients

(kinematic viscosity, thermal and concentration coefficients) are assumed

4
to be constant. The tangential velocity ve imposed is

where T is the circulation of the vortex, i/ is the kinematic viscosity and

Tf — I

is a similarity variable.

With the assumptions described above, the equations for species and

energy are decoupled from the momentum and continuity equations; they

are equations representing a balance between convection, diffusion and re- action. The reaction rate, which is very nonlinear in concentrations and

temperature in general, can be eliminated by taking a linear combination of

the dependent variables when the additional assumption is made that the

thermal and species diffusion coefficients are equal. This is a common procedure in the theoretical analysis of combustion equations [2]. The linear

combinations are often called coupling functions or Shvab-Zeldovich variables, and in this case all satisfy the same equation; therefore, one need

only solve for one dependent variable.

The Shvab-Zeldovich variable Z satisfies the convection-diffusion equation

dt ^ r dO \ dr^ ^ r dr J

where D is the species diffusion coefficient, assumed to be constant and the

same for each species. The initial conditions are that Z = 1 for tt/2 < ^ <

37t/2 and Z = 0 for —tt/2 < 0 < tt/2.

Integrating the tangential velocity gives the angle 0(r, dQ^i) at time t for

any fluid element initially located at r,9o . A change of variables to the

Lagrangian coordinates, p,do,T,

r =

$ =

t =

where Ej{z) = exp(—zt)dt, can then be made in the equation for Z.

Finally, assuming that Z is only a function of the similarity variable rj and

Oq +

r l-E2{ri)

Sttu rj (3)

5
the angle and performing a Fourier decomposition in the angle,

oo = X] Zn(rj) exp(indo) (4)

a system of equations for each of the Fourier mode amplitudes is obtained:

where

r r \ f \ rw r\

/ X ^ 1 . 1 ~ exp(— r?) ^

}„{n) = Sc H h in T Re

(5)

9r,(n) = -

in ex

1 - exp(-77)

Re + 1

P(-'/) - [(1 - exp(-r/))/r;]

4r/2

Re

and where Sc = i/jD is the Schmidt number and Re = T/^ni/ is the

Reynolds number based upon the circulation F.

This equation is an ordinary differential equation for in the similarity

variable 77 . Boundary conditions are that the solution remain bounded as

77 goes to zero and that the initial conditions on Zn are recovered as 77 goes

to infinity. From the symmetry of the problem, the initial conditions are found to be, for even values of 71, Zn(v —> c») = 0, and for odd values of

71 = 2771 -h 1,

(6 ) For computation, the infinite endpoint is replaced by a finite parameter.

A very special case discussed in more detail in Section 3.2 can be used

to test the numerical methodology and the computer programming. It also

allows us to determine the boundary conditions in the similarity variable

from initial conditions in the original variables. It is the pure diffusion case,

which arises when the circulation F is taken to be zero. The known solution

6
for the diffusion of two half-spaces into each other is given in terms of an

error function, which is expressed in cylindrical coordinates and Fourier

analyzed to give in terms of Bessel functions. This result has been used

to examine the sensitivity of the numerical solution to the finite truncation

of the infinite interval, to the number of terms retained in the Fourier

synthesis of Z, and to the mesh size used in the solution of the two point

boundary value problem, Eq.(5).

The general solution for Zn(rj) is a- complex-valued function of the real

variable rj. From the governing equation, it is seen that Z-nM is the

complex conjugate of Zn(rj). The modes Z„ are synthesized using an FFT

routine to determine Z(r),0o), and information about the location of the

flame-sheet and the rate at which fuel is consumed can be determined.

3 Solutions

3.1 Small Circulation

When Re = 0, the problem reduces to the simple one of a diffusing, reacting

interface; fuel in the left half-space and oxidizer in the right half-space

diffuse into each other and react. For this problem, the Shvab-Zeldovich

variable Zo(r}^$o) is given by = (1/2) erfc (^Scr/cos^o) (7)

In this case the Eulerian and Lagrangian coordinates are the same. Fourier

analysis yields

oo Zo(r)Jo) = Ao-h 2 A2m+i(^)cos[(2rn + l)^o]

TM=0

where Aq = 1/2 and

( 8)

v 27T 2m + 1 [7^(Scr//2) + -fm+l (Sc 77 /2)] The amplitudes A2 m+i('n) satisfy

(9)

7
I / cj ^^dA2m+l (2t71 -|-

( 10) Compare Eq.(5) with Re = 0 and n = 2m -f 1. The symmetry of the

problem implies that all even Fourier modes are zero. Boundary conditions for Eq.(lO) are determined from the Fourier ex- pansion of the diffusion solution when r; ^ oo. These boundary conditions, are simply the initial conditions for the pure diffusion problem; for a finite

value of radius rj oo a.s t ^

0. Hence in Eq.(7) for 0 < ^ 27t

Z{ri oo, ^o) = ^(^0 - 7t/2) - H(0o - 37t/2) (11)

where H(x) is the Heaviside function of x. Fourier analysis of Z(oo,^o)

provides the boundary values for A2 m+i(v) as 77 —» oo : Aq = 1/2 and

A,^Mn-^oo)=

^ ( 12)

Note that the boundary conditions for the pure diffusion problem in terms

of the similarity variable are also the boundary conditions for the general

Marble problem, since the initial conditions for both problems are the same. The solution given by Eq.( 8) and Eq.(9) have been used cis the base

state for a perturbation analysis when the Reynolds number is small. Only

odd modes need be considered, and since Fourier modes of negative integer

n are the complex conjugate of the corresponding Fourier mode of positive

integer, only Fourier components with positive integers need be considered. Comparison between results obtained by this technique and results

obtained by direct numerical computation of the mode amplitudes using a

tridiagonal solver affords a useful check on the numerical procedure used

to solve Eq.(5). We have achieved agreement to three significant figures for

Re =0.1 and Sc=l.

3.2 Large Schmidt Number

When the Schmidt number is infinite, diffusion is unimportant and Eq.(2) is purely a convection equation in which the change of variables to Lagrangian

8
coordinates reduces the problem to a trivial one. When the Schmidt number is large, asymptotic methods allow one to determine an approximate

analytical solution to Eq.(5) from which the character of the solution to

Eq.(2) for large Reynolds number can be determined. A change of dependent variable to Wn(r)) where

Zn{n) = exp (-Sc 77/2 -In 77/2- (m Re / 2) [£'2 (77) - 1]/t7) ( 13) gives the equation

where

+ £^(77; Re , Sc )W„(t7) = 0 ( 14)

Fn(^; Re, Sc) =

-(nV2-l)

2772

(Sc + 1/77)^ mRe Sc (1 — e ’’) , . ri

A singular perturbation analysis of Eq.( 14) for large Sc yields two so- lutions W^[r]) and ^“(77), of which the former grows exponentially for

large 77, and the latter decays. The solutions, in turn, imply that the corre- sponding solutions Z^(t7), Z~ (77) become constant and decay exponentially

for large 77. Since the desired solution to Eq.(2) must pass to the purediffusion case for large 77, we have- 00)

^

and the constant multiple of -^^^+i(^) is determined. Since decays

exponentially as 77 —> 00, any multiple of Z^m+iM added and we still have a solution. However, the solution for Z2m+i(^) must be bounded as

77 —> 0 , and this fixes a linear combination of Z^m+i Examination

of Z2m+i to leading order in Sc shows that it is a constant throughout while

the other solution is singular at 77 = 0 in the limit as Sc -^00. Hence it is the desired solution!

Carrying out the asymptotic analysis for large Sc to next order yields — (2771 -H 1 )^

4 Sc 77 (A - iB) ( 17)

9
where

A

B

1 +

Re

2 Re

2m + 1

- - 2E4r,) + E4(2r,)

^^

+ E4n)

2rj rj Two observations can be made about this asymptotic expression. First,, as Sc —S' oo, the argument of the exponential goes to zero and

goes to the appropriate constant, namely ( — l)”^'''^/[27r(2m + 1)]. Second,

for small rj, goes to zero as exp [— const. /rj].

Substitution of Eq.(17) into Eq.(4) and performing some simplification

yields the approximate solution to Eq.(2) for large Schmidt number:

Z{n,eo) = 1/2 + 2 f;

I sin[(2m + l)$ol (18)

where

Am(rr, Re , Sc ) = (2m + 1) 4 oc 77 _ . 7T Re ~ / V / \ ^0 = T

- - ^Mn) (19)

and where

7i(h) = |l/3 - 2E4ri) + Ei{2i})\lr]‘^

?2(n) = l(.S3(h) - l/2)/»? + E4r})\h

Plots of functions fi and /2 are shown in Figures 2a and 2b; they are rather smooth functions which allow us to determine the analytical behavior

of Z for large Schmidt numbers. For small ry, /i —»> 1 and /2 —> (l/2)lnr7. For large rj, fi and J2 -1/(277).

Some important and illuminating observations can be made from the

asymptotic solution Eq.(l8). First, note that Z — 1/2 is a Fourier sine series. When the sum of the terms in this series is small, the solution for the

Shvab- Zeldovich variable is approximately 1/2. Since, for a stoichiometric

mixture, Z = 1/2 is the equation for the flame sheet, the flame sheet for

10
such a mixture occurs near where the sum of the Fourier series is small.

Now, all terms of the series are exactly zero when = 0? and this condition determines the equation for the flame sheet ^o(^) for a stoichiometric

mixture. However, the terms in the series will also be small when the argument of the exponential is large, and, when the Reynolds number is large,

this occurs provided rj is not too large.

The physical interpretation of these mathematical statements for large

Reynolds and Schmidt numbers is as follows. For a stoichiometric mixture,

there are two reaction regions. In the outer region there is a flame sheet,

which remains close to the convectively mixed interface in the absence of

diffusion. This interface is determined by the equation = 0, and for

moderately large rj, is very close to do = ±7r/2, which are the equations for

the initial interface in the Lagrangian coordinate system. In the inner re- gion, there is a burnt core in which both fuel and oxidizer are depleted, and

the growth of this core is determined by the condition that the arguments

of the exponentials are large enough that each of the terms in the Fourier

series is negligible. For a large Reynolds number (and Schmidt number)

this condition determines a value of the similarity variable, rj* say, and the

growth of the burnt core is determined then by the equation /4iyt = r/*. The observations about the burnt core are consistent with those made by

Marble and Karagozian [14], [23], [24].

3.3 Special Cases

Two special cases provide insight into the interplay between the analysis

and the physics and give limiting cases against which to check calculated

results. The flrst cztse is that for which a solution is provided in Subsection

3.1, namely the pure diffusion case. Here, no vorticity is present so that

r = 0 and Re = 0. In Figure 3 the solution for this case is shown; contours

of constant amplitude Z are shown in either a Lagrangiaji or an Eulerian

coordinate system, since when Re = 0 both coordinate systems are the

same (essentially do and the square root of 77). The plot is not shown all the way to the origin for technical reasons and only goes out to the square

root of ten, which is related to the value used to truncate the inflnite

interval in the similarity variable. The flame sheet is located along any one

11
of these constant-amplitude contours, depending upon the concentrations

of fuel and oxidizer initially in each half space; for a stoichiometric mixture

initially, the contour Z = 1/2 is where the flame sheet resides. Since this case is for pure diffusion, the contours should be straight lines (distorted

here by the plotting routines); Z varies in a one-dimensional manner from

its value of one in the left half-plane to zero in the right half-plane. The

contours for which Z > 1/2 are shown as solid lines, whereas the countours

for which Z < 1/2 are shown as dashed lines.

The second simple case to calculate and to physically interpret is that

in which there is no diffusion, i.e., = 0 and therefore Sc = oo. In

this case Eq.(2) becomes purely a convective equation, which is solved by

the Lagrangian to Eulerian coordinate system transformation, Eq.(4). In

Figure 4 are shown plots of the interface for four values of the Reynolds

number Re = r/(47n/) = 2,4,6 and 8. Two points should be emphasized

about this figure. First, these plots do not represent the time evolution or convective mixing of a particular instance of the problem, even though the

sequence gives this illusion. Second, the time evolution of any one example

of the problem, i.e., one Reynolds number, can be visualized as follows.

The interface shape is a curve or functional relation between B and rj, and

any location on this interface is given by a specific pair rj,0. Since rj -

r^/4i/t, a particular location on the interface at one time t will determine

a radial position r. At a later time, the interface shape does not change

and this location will “diffuse” out to a new radial position determined by

the condition that rj remains constant. Therefore, the interface will retain

the same shape, determined by the rj^B relation as time increases, but the

length scale characterizing the interface will increase as r oc \/t.

3o4 Numerical Results

The solution given by Eq.(7) and that given by Eqs.(8) and (9) are for

pure diffusion and can test the numerical methodology used to solve the

ordinary differential equations (5) and the FFT procedures needed for the

solution for Z(rj,Bo). Numerical solution of Eq.(5) has been performed for

many values of the parameters Re and Sc using a central difference approximation to the differential operators. The interval of integration for

12
the differential equation is truncated to carry out the numerical integration and asymptotic values for the Fourier amplitudes are applied at the

truncated location. A finite number of Fourier modes has been computed,

and, ELS noted above, only odd modes of integer greater than zero have been

computed. The analytical behavior of the solution for small values of the

independent variable rj must be handled carefully for lower order modes;

therefore, a different dependent variable is computed near the origin. The

pure diffusion case, when Re = 0, has been used to assess accuracy, particularly the effects of the finite truncation value, the number of Fourier modes

and the number of grid points in the finite difference scheme required for a

specified resolution.

In Figure 5 there are four mode amplitudes displayed, modes 1, 3, 5, and

7, ELS functions of the similarity variable rj for Reynolds number zero and

Schmidt number one. Mode 1 is in the upper left corner, mode 3 is in the

upper right corner, mode 5 is in the lower left corner and mode 7 is in the

lower right corner. In each case the solid line is the real part of the mode

amplitude, the dashed line is the imaginary part and the asterisks line is the modulus. Since this is the pure diffusion case, the amplitudes are real

functions. For this calculation, 99 integration nodes and 15 Fourier modes

were used, and the dependent variable was changed at r? = 1. Each of the

amplitudes should asymptote for large values of the independent variable rj; the truncation value of rj has been taken as ten. From the figure it can be

seen that modes 1,3 and 5 seem to be asymptoting, but that mode 7 is still growing at r; = 10. However, it should also be noted that the magnitude

of mode 7 is at least an order of magnitude smaller than that of mode 1. In Figure 3, shown in the previous subsection as the special case of pure

diffusion, the Fourier synthesis is shown of the modes determined from the

computations described in the preceding paragraph. As noted before, the

plot is shown in a coordinate system representing either the Lagrangian or Eulerian systems, since they are the same when the Reynolds number is zero.In Figure 6 the Fourier amplitudes for modes 1, 3, 5 and 7 are shown

in the same order as in Figure 5, but for a Reynolds number of 2 and all else the same as described for the calculation shown in Figure 3. The first

difference to note is that the Fourier amplitudes are complex in this case.

13
Also, the mode amplitudes are developing an oscillatory nature for small

values of 77 , in contrast to the case of zero Reynolds number.

In Figure 7 is shown the Fourier synthesis in Lagrangian coordinates of

the mode amplitudes from the calculation shown in the previous figure. The

fiame sheet has been designated to be at Z — 1/2 and contours of constant

Z for Z > \ j2 are shown as solid lines whereas contours of constant Z for

Z < 1/2 are dashed lines. Note that the contours approach straight linesaway from the origin, but close to the origin diffusion is winding up the

contours in the direction opposite to the rotation (the rotation due to the

vortex is counterclockwise).

In Figure 8 is shown the corresponding plot of constant- Z contours in

the Eulerian coordinate system. It is interesting to note that the contours

are distorted by the convective winding of the vortex, but that near the

origin, there is much less winding than in the Lagrangian coordinate system.

3c5 Large-Schmidt-Number Analytical Results

The numerical computations become increasingly difficult as the Reynolds

number is raised both because the convective mixing becomes more convoluted within a specified radius and because the radial distance over which

the calculations must be carried for a particular accuracy must be substantially increased. Therefore, the large Schmidt number analysis is very

useful for examining the structure of the solution for large Reynolds number. Again, this is also a quite realistic assumption for diffusion-controlled

reactions in miscible liquids.

In Figure 9 is shown plots of the interface shape calculated from the

large Schmidt number analysis for three values of the Reynolds number,

1, 10 and 100, in a Lagrangian coordinate system; these plots have been

calculated for a Schmidt number of 10, which satisfies the requirements

of the analysis presented in Subsection 3.2 that the Schmidt number be

large. These plots have the same scale, so that structures formed at larger

Reynolds number extend to larger radius (or similarity variable rj). For

Reynolds number of unity, in Lagrangian coordinates, the interface hardly

varies from a straight line. The circle around the origin shows the radius

at which the burnt core is located (the location at which the amplitudes in

14
Eq.(18) are down from the value 1/2 by exp (—3)). The two plots for larger

Reynolds numbers, 10 and 100, show increased distortion of the interface

around the origin and growth of the circle representing the burnt core. The distortion of the interface is in the direction opposite to that of the

convective mixing, which is positive in the counterclockwise direction, as noted in the discussion of the numerical results in the last subsection. As

noted there, this is because the mixing increases the gradients between fuel

and oxidizer, enhancing counterrotation diffusion. Note furthermore that

the deviations of the interface from planar outside of the burnt core is very

small, showing that beyond the burnt core the interface between fuel and

oxidizer is essentially controlled by the convective mixing.

In Figure 10 is shown plots of the interface shape in Eulerian coordinates for the same three values of Reynolds number and for the Schmidt

number shown in Figure 9. These plots are drawn approximately to scale, so that structures formed at larger Reynolds number extend to larger ra- dius (or similarity variable rj). In each plot the area has been blackened

within the burnt core to indicate no reaction activity. As in Figure 9, the

burnt core grows with increasing Reynolds number. Once again it should

be emphasized that these plots do not represent time evolution, but rather

different parametric configurations; time evolution is according to the similarity laws determined by the similarity variable 77 = r^/47rt. The amount

of convective mixing in the counterclockwise direction increases dramatically with Reynolds number; both the radial distance from the origin at

which the interface deviates from planar and the winding of the interface

inside of this radial location are substantially larger for Re = 100 than for

Re = 10.

4 Consumption Rate

Important quantities of interest from this analysis are the global consumption rates for fuel and oxidizer and the global rate of heat release; it is desired to calculate these quantities as functions of the Reynolds number,

Schmidt number and initial concentrations of fuel and oxidizer. The con- sumption rates under general conditions can be calculated from the analysis

presented earlier but require computation of Fourier amplitudes using an

15
ODE solver, synthesis of the solution using FFT routines, location of the

flame surface using a root flnder and integration over the whole sheet to

obtain the global rates.

The local rates of consumption of fuel and of oxidizer and the rate

of heat release are all proportional to the derivative of Z normal to the

flame surface. Therefore, the calculation of these quantities requires us

first to locate the flame sheet and then to determine the derivative of the

Z normal to it. When the Schmidt number is large, the analytical results

of Subsection 3.2 can be used both to locate the flame sheet and to obtain

an expression for the normal derivative of Z. When the initial mixture of

fuel and oxidizer is stoichiometric, the flame sheet occurs at Z = 1/2 and is located along the curves determined in Lagrangian coordinates by = 0

(Eqs.(19)):

^ 7T Re ~ , V , .

These lines are transformed by Eqs.(4) into the Eulerian system:

e = + ^Mn) +

^

[1 - E,(n)\ (21)

The flame sheet then consists of two curves each of the form 0(r)).

The tangent and the normal to the curve defining the flame sheet are given by the expressions

where, from Eq.(2l),

dO

r —

dr

Rer/ —

drj

hM

Sc

+

1 - E2(v)

or d0 Re _

dr I Sc

1 - 2E3(n) 2E2{ri)

V

-E,{n)

1 - E^iri)

+ Ei(ri)

(22)

(23)

(24)

16
The gradient of Z in cylindrical coordinates, irdZjdr + (islr)dZjdO^

can be found in Lagrangian coordinates using Eq.(4) and then in terms of

the similarity variables 77 = r^/ (4i/t),9o (where we note that the Lagrangian

variables p and t can be replaced by r and t):

VZ = -

r l-exp(-n)dZ

^ W,

- 1 azr

tr H

rdO (25)

0 The analytical expression Eq.(l8) for Z can be differentiated and evaluated

along the flame sheet ^0 = 0- When this is done, we And

drj

dZ

Wo

Re df2 \ dZ

2 Sc dr) ) dOo = ±-2

^

2^ exp

7T m=0

(2m + 1 )' 4 Sc 77 1 +

Finally, the consumption rates of fuel and oxidizer locally are

dC = D^ds = Dn VZds

dn

(26)

(27)

where D is the species diffusion coefficient, n is the normal direction to the

flame sheet, and s is the arc length along the flame sheet. Here

ds = 1 + (28)

Then the total consumption rate for either species is C = D [ ~ds (29)

J flamesheet dfl

One additional complication arises at this point. In the case when there

is no vortex imposed on the reaction process, i.e. in the pure diffusion case,

the consumption rate is inflnite because there is an infinitely long flame

length over which the reaction takes place. When the vortex is imposed,

the reaction rate is still inflnite because there is also an inflnite flame length

beyond the region influenced by the vorticity. Therefore, we determine the

17
enhancement caused by the imposition of the vortex. If we denote by Co

the consumption rate in the pure diffusion case (Re =0), then we desire

C-Co.

Calculation of the consumption rate enhancement at this point would

require numerical integration. In an attempt to push the analytical calculations as far as possible and to obtain expressions with which to compare

with those derived by Marble and Karagozian [14], [23] [24], we furtherassume that the Reynolds number is large. A large Reynolds number is one for which the functions Ami’n'i Rc, Sc) appearing in the exponential

amplitudes in the asymptotic solution Eq.(l8) are large enough that these

amplitudes are small; then the terms in the Fourier series decrease very

rapidly with increasing m and for our purposes, the Fourier series Ccin be

approximated by only one term. Therefore, for m = 1, we require that

Ai > 3 approximately; a Reynolds number of 100 or more is adequate for

example. Then, the asymptotic expressions for fi and /2 for large ry ,which

were given earlier, can be used. Using only the first term in the Fourier

series solution Eq.(18) and taking only the “large ry” expression for the

equation for the flame sheet and for the Eulerian-Lagrangian transformation, we find

dO Re / 1 r — « ( 1 dr ^ V 2 Sc

and

1— exp

7T

1 / Re^ \

4 Sc ?y y ^ 3/y2 j

dr} (31)

Integration of the approximate expression for the local enhancement of

the consumption rate over the flame sheet, then gives the global enhanced

species consumption rate:

^ ^ f(2/3) „ /'2Re^Sc^V'^^ C - Co « -' —D (32)

and the parametric dependence of this expression agrees with that reported

by Marble and Karagozian [14], [23], [24].

18
5 Conclusions

As noted earlier, this problem was originally formulated by F. Marble and

analyzed by Marble and Karagozian [14], [23] using solutions developed by

Carrier et al. [15], which are valid locally along the flame sheet. In this

paper we analyze the same problem by alternate methods, ones that accurately solve the equations globally. In particular, the Marble problem has

no natural length nor time scale associated with it, and, therefore, allows a global similarity solution, an observation which reduces the number of

dependent variables from three (radius, angle and time) to two (angle and

the similarity variable - which has the character of similarity variables for

diffusion processes). Perhaps the most important analytical result is that

which expresses the dependence of the species consumption rate on the

Reynolds and Schmidt numbers for large values of these parameters, and

this result corroborates that found by Marble [14].

The solution presented here elucidates, we believe, the nature of the

problem, and the method of solution provides all of the machinery necessary

to calulate additional results to whatever accuracy one might desire for

other purposes, such as comparing with more general numerical solutions

[16]- [21]. The key features of the analysis are the observations that the

problem permits a global similarity solution and that the equations can

be transformed to Lagrangian coordinates. The first observation reduces

the number of independent variables from three to two, and the second

eliminates scaling difficulties arising in convection-diffusion equations when

the Reynolds number is large. The Marble problem is inherently interesting

because it addresses the question of enhancement of species consumption

and heat release rates by flame stretching in a simple geometry. It is also

of interest because, as discussed earlier, it simulates the enhancement in

these rates in a vortex generated in a two dimensional shear layer [9]-[l2].

The Marble problem can clearly be extended in several directions. Obvious directions for generalization are to consider other flow fields, more

general geometries for the initial fuel and oxidizer conflgurations, and the

effects of density changes induced by heat release on mixing. A more general flow field and density variations have been considered analytically by

Karagozian and Marble [23], [24], using the same type of local analysis orig19
inally employed by Marble. A more general flow and generalized species

geometries are investigated analytically in the paper by Baum et al. [l3j; a three dimensional flow fleld consisting of a stretched vortex is considered, a very important generalization for turbulence where vortex stretching is crucial. Finite regions of fuel in an oxidizing atmosphere, e.g. a sphere of

fuel in an oxidizing background, are also a practical generalization considered in the paper of Baum et al. These generalizations are also the subject

of investigation by numerical techniques [l6]-[2l].

Finally, the work presented here can be viewed in a larger context,

where it, or its generalization, is regarded as a submodel of a more general

model for turbulent combustion. The large scale fluid motion for whatever

geometry must be computed in a large Reynolds number or essentially

inviscid approximation. We have developed such a large scale model for

buoyant convection induced by a room Are [25], [26], [27]. However, the

large scale flow could as well be calculated for other conflgurations, and

then the small scale combustion submodel would be embedded into this

flow fleld.

We wish to thank Dr. G.B. McFadden for a useful discussion and commments on the paper and Dr. C. Fenimore for helpful comments on the

paper. This research was partially supported by the Air Force Office of

Scientiflc Research under Contract AFOSR-ISSA-87-0018.

References

[1] Libby, P.A. and Williams, F.A. (eds). Turbulent Reacting Flows, SpringerVerlag, Berlin, 1980.

[2] Williams, F.A., Combustion Theory, The Fundamental Theory of Chemically Reacting Flow Systems, Second Edition, The Benjamin/Cummings

Publishing Co., Inc., Menlo Park, California, 1985.

[3] Buckmaster, J.D. and Ludford, G.S.S., Theory of Laminar Flames,

Cambridge University Press, 1982.

[4] BuckmcLster, J.D. and Ludford, G.S.S., Lectures on Mathematical Combustion, Society for Industrial and Applied Mathematics, Philadelphia,

20
Pennsylvania, 1983.

[5] Oran, E.S. and Boris, J.P., Detailed Modelling of Combustion Systems,

Prog. Energy Combust. Sci., Vol. 7, 1981, pp 1-72.

[6] Peters, N., “Laminar Diffusion Flamelet Models in Non-Premixed Turbulent Combustion”, Prog. Energy Combust. Sci., Vol. 10, 1984, pp

319-339.

[7] Oppenheim, A.K., “The Beauty of Combustion Fields and Their Aerothermodynamic Significance”, Dynamics of Reactive Systems Part I: Flames

and Configurations, J.R. Bowen, J.-C. Leyer and R.L Soloukhin (eds).

Progress in Astronautics and Aeronautics, Vol 105, 1986, A.I.A.A.,

1633 Broadway, N.Y.,N.Y. 10019.

[8] Ashurst, W.T., “Vortex Simulation of Unsteady Wrinkled Laminar

Flames”, Combust. Sci. Technology, Vol. 52, 1987, p 325.

[9] Brown, G.L. and Roshko, A., “On Density Effects and Large Structure

in Turbulent Mixing Layers”, J. Fluid Mech., Vol. 91, 1974, pp 319-

335.

[10] Roshko, A., “Structures of Turbulent Shear Flows: A New Look”,

A. LA.A. Journal, Vol. 14, 1976, pp 1349-1357.

[11] Mungal, M.G. and Dimotakis, P.E., “Mixing and Combustion with

Low Heat Release in a Turbulent Shear Layer”, J. Fluid Mech., Vol.

148, 1984, pp 349-382.

[l2j Browand, F.K., “The Structure of the Turbulent Mixing Layer”, Physica Vol 18D, 1986, pp 135-148.

[13] Baum, H.R., Corley, D.M. and Rehm, R.G. “Time-Dependent Simulation of Small-Scale Turbulent Mixing and Reaction”, Paper presented

at the Twenty-First International Symposium on Combustion, Munich, West Germany, August 3-8, 1986.

[14] Marble, F.E. “Growth of a Diffusion Flame in the Field of a Vortex”,

Recent Advances in Aerospace Sciences (C. Cassci, Ed.) 1985, p.315.

21
[15] Carrier, G.F., Fendell, F.E. and Marble, F.E., “The Effect of Strain

Rate on Diffusion Flames”, SIAM J. Appl. Math., Vol.28, No. 2,

March, 1975.

[16] Riley, J.J., Metcalfe, R.W. and Orszag, S.A., “Direct Numerical Simulations of Chemically Reacting Mixing Layers” , Phys. Fluids, Vol. 29,

1986, pp 406-422.

[17] McMurtry, P.A., Jou, W.-H., Riley, J.J. and Metcalfe, R.W., “Direct

Numerical Simulations of a Reacting Mixing Layer with Chemical Heat

Release”, A. I. A. A. Journal, Vol 24, 1986, pp 962-970.

[18] McMurtry, P.A. and Riley, J.J., “Mechanisms by Which Heat Release Affects the Flow Field in a Chemically Reacting, Turbulent Mixing Layer”, Paper AIAA-87-0131, paper presented at the AIAA 25th

Aerospace Sciences Meeting, January 12-15, 1987, Reno, Nevada.

[19] McMurtry, P.A., “Direct Numerical Simulations of a Reacting Mixing

Layer with Chemical Heat Release”, Ph.D. Thesis, Mechanical Engineering, University of Washington, Seattle, Washington, 1987.

[20] Ghoniem, A.F. and Givi, P., “Vortex-Scalar Element Calculations of a

Diffusion Flame”, Paper AIAA-87-0225, paper presented at the AIAA

25th Aerospace Sciences Meeting, January 12-15, 1987, Reno, Nevada.

[21] Laverdant, A.M. and Candel, S.M. “A Numerical Analysis of a Diffusion Flame-Vortex Interaction”, Paper presented at the SIAM Conference on Numerical Combustion, San Francisco, California, March

9-11, 1987.

[22] Rehm, R.G., Baum, H.R. and Lozier, D.W. “Two-Dimensional Flame

in a Vortex Field”, Paper presented at the SIAM 1986 National Meeting, Boston, Mass. July 21-25, 1986.

[23] Karagozian, A.R. and Marble, F.E., “Study of a Diffusion Flame in a Stretched Vortex” , Comb. Sci. Tech., Yo\. 45, 1986.

22
[24] Karagozian, A.R., “An Analytical Study of Diffusion Flames in Vortex

Structures” , Ph.D. Thesis, California Institute of Technology, Pasadena,

Cal., 1982.

[25] Rehm, R.G. and Baum, H.R., “The Equations of Motion for Thermally

Driven, Buoyant Flows”, J. Research Nat, Bur. Standards., VoL 83,

1978, pp 297-308.

[26] Baum, H.R., Rehm, R.G., Barnett, P.D. and Corley, D.M., “ Finite

Difference Calculations of Buoyant Convection in an Enclosure, I. The

Bzisic Algorithm”, SIAM J. Sci. Stat. Comput., Vol. 4, 1983, pp

117-135.

[27] Baum, H.R. and Rehm, R.G., “Calculations of Three Dimensional

Buoyant Plumes in Enclosures”, Comb. Sci. Tech., Vol. 40, 1984, pp

55-77.

23
Figure Captions

1. Schematic diagram of the Marble problem. Fuel, in the shaded left halfplane, and oxidizer in the right are allowed to react at the thin flame sheet separating them. Simultaneously, a line vortex with its axis at the origin induces

convective mixing between the two species. The objective is to calculate the en- hancement of the species consumption caused by the mixing.

2a. Function fi(r}) deflned following Eq.(19); the function arises in the large

Schmidt number asymptotic analysis.

2b. Function /2 (^) defined following Eq.(l9); the function arises in the large

Schmidt number asymptotic analysis.

3. Contours of constant species concentration (constant Z) for pure diffusion. In this special case the vorticity is zero so that F = 0 and Re = 0, and,

therefore, there is no convective mixing. The contours should be straight lines

(distorted by the plotting routines) with the species concentrations changing in a one-dimensional fashion from the initial concentration of fuel on the left to the

initial concentration of oxidizer on the right. The flame sheet could be located

along any one of these contours depending upon these initial concentrations; in

this plot a stoichiometric mixture existed initially and the flame sheet occurs along

the initial interface with constant-density fuel contours shown by solid lines and

constant-density oxidizer lines dashed. Contours are not shown all of the way to

the origin, and a horizontal solid line is shown for technical reasons.

4. The interface between fuel and oxidizer for four values of the Reynolds number, Re =2, 4, 6 and 8, when there is no diffusion (D = 0 and Sc = oo): Reynolds

number increases from top to bottom. These plots illustrate the convective mixing

induced by the vorticity and show that the mixing is enhanced as the Reynolds

number is increased.

5. Numerical solutions to Eqs.(5) for four mode amplitudes, modes 1, 3, 5 and

7, as functions of the similarity variable r} for Reynolds number zero and Schmidt

number one and a stoichiometric mixture. Mode 1 is in the upper left corner, mode 3 in the upper right, mode 5 in the lower left and mode 7 in the lower

right. The solid curve is the real part of the mode amplitude, the dashed curve is the imaginary part and the asterisks curve is the modulus. Since this is the pure

diff'usion case, the amplitudes are real. There were 99 integration nodes used for

the calculation.

6. Numerical solutions to Eqs.(5) for four mode amplitudes, modes 1, 3, 5 and

7, as functions of the similarity variable rj for Reynolds number two and Schmidt

24
number one and a stoichiometric mixture. All other conditions are the same as those in Figure 5.

7. Solution of Eq.(2) in the Lagrangian coordinate system for a stoichiometric

mixture, a Reynolds number of two and a Schmidt number of unity. This solution

is found using Fourier synthesis of 15 modes of which four are shown in Figure

6. Contours of constant fuel density are shown as solid curves while contours

of constant oxidizer are dashed; the flame sheet occurs at the solid curve which

divides fuel and oxidizer regions. Note that the contours approach straight lines

away from the origin, and close to the origin, diffusion induces counterclockwise

gradients (opposite to the clockwise mixing of the vortex).

8. Solution of Eq.(2) in the Eulerian coordinate system for the case shown in

Figure 7. In this coordinate system, the contours are distorted by the convective

winding of the vortex, but near the origin, there is much less winding than in the

Lagrangian system.

9. Plots of the interface shape for three values of the Reynolds number, 1, 10

and 100, in a Lagrangian coordinate system as determined by the large Schmidt

number asymptotic analysis: these plots were calculated for a Schmidt number of

10, which satisfies the requirements of this analysis. The circle around the origin

shows the radius at which the burnt core is located as discussed in the text.

10. Plots of the interface shape in Eulerian coordinates for the same conditions

shown in Figure 9. In each plot, the area within the burnt core has been blackened.

The amount of convective mixing and the distance from the origin at which the

interface deviates from planar increases dramatically with Reynolds number.

25
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*
Vortex, Flame-Sheet

Interaction

Fuel

t|A

1A

Flame Sheet

Oxidizer

V

Figure 1

F2=(

(E3(ETfi)-0.5)/ETfl+E2(ETR)

)/ETfl

o

NOIlGNrU

n

CN

Q)

u\

mLL

CO . » o (0 o • o

Eh iH 05 iH iH < 1 < O 1 1 1 1 l-C 1—

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Figure 4

90‘
Figure 7
Figure 8
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Figure 9 o

tH

tH

Schmidt

No.
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System

Figure

10

Schmidt

No.
J!NBS-n4A (REV. 2»ac)

j U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date

1 BIBLIOGRAPHIC DATA

1 SHEET (See instructions) REPORT NO.

NBSIR- 87-3572 JUNE 1987

j 4. TITLE AND SUBTITLE

.Diffusion-Controlled Reaction in a Vortex Field

5. AUTHOR(S)

ReGo Rehm, HoR. Baum, and D.W. Lozier

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Gramt No. AFOSR-155A-87-0018

8. Type of Report & Period Covered

9/86-9/87

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, State, ZIP)

(Partially supported by)

Air Force Office of Scientific Research

Bolling Air Force Base

Washington, DC 20332

10.

SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200~word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

A two-ddLmensional model of a constant-density diffusion controlled reaction between

unmixed species initially occupying adjacent half-spaces is formulated and analyzed.

An axisymmetric viscous vortex field satisfying the Navier-Stokes equations winds up

the interface between the species as they diffuse together and react, A flame-sheet

approximation of the rapid reaction is made using Shvab-Zeldovich dependent variables

The model was originally proposed by F» Marble, who performed a local analysis and

determined the total consumption rate along the flame sheet. The present paper

describes a global similarity solution to the problem which is Fourier analyzed in a Lagrangian coordinate system. An asymptotic analysis of the Fourier amplitudes, valid

for large Schmidt numbers is presented. The solution is evaluated numerically in

Lagrangian and Eulerian coordinate systems. This problem has been studied as part of

a more general model which has application to the description of turbulent combustion.

12.

KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) combustion; convection; diffusion-controlled; flame-sheet; reaction; turbulent mixing

1.3. AVAILABILITY

Cxi Unlimited

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